| 一类KdV-Burgers方程的奇摄动解与孤子解 |
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| 引用本文： | 包立平,李瑞翔,吴立群.一类KdV-Burgers方程的奇摄动解与孤子解[J].应用数学和力学,2021,42(9):948-957. |
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| 作者姓名： | 包立平 李瑞翔 吴立群 |
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| 作者单位： | 1. 杭州电子科技大学 理学院, 杭州 310018; |
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| 基金项目： | 国家自然科学基金(51775154)；浙江省重点自然科学基金（LZ15E050004） |
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| 摘 要： | 讨论了一类具有大Reynolds数且弱频散性的KdV-Burgers方程, 在数学上表示为一类奇摄动KdV-Burgers方程.KdV-Burgers方程中含有的非线性项与频散项互补作用形成稳定向前传播的孤立子.通过数学分析, 描述了孤立子的传播途径和传播速度等物理量的发展变化规律.通过奇摄动展开方法, 构造了该问题的渐近解.首先，用Riemann-Earnshaw方法求得退化解, 得到了简单波, 该简单波波形中的任意一点与初始点都存在一个传播速度差, 这使得波在传播过程中波形不断畸变, 最终形成冲击波面, 即间断面, 在它的两侧质点的速度有一个跳跃, 且随时间不断变化；其次, 在退化解的间断曲面处做变量替换, 构造一种修正的行波变换, 得到了内解展开式的孤子解, 并证明了内外解的存在性与唯一性；最后,通过一致有界逆算子的存在性做了余项估计, 并得到渐近解的一致有效性.结果表明, KdV-Burgers方程在大Reynolds数且弱频散性的性质下,扰动集中在退化解的间断面附近,孤立子链接两侧质点,其传播途径不是时间与空间的线性形式,而是沿着退化解的间断面附近传播,形成稳定的波形.
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| 关 键 词： | KdV-Burgers方程 弱频散性 孤子解 奇摄动 一致有效性 |
| 收稿时间： | 2021-01-21 |
Singularly Perturbed and Soliton Solutions to a Class of KdV-Burgers Equations |
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| Affiliation: | 1. School of Sciences, Hangzhou Dianzi University, Hangzhou 310018, P.R.China;2. School of Mechanical Engineering, Hangzhou Dianzi University, Hangzhou 310018, P.R.China |
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| Abstract: | A class of KdV-Burgers equations with large Reynolds numbers and weak dispersions were discussed, which were mathematically expressed as a class of singularly perturbed KdV-Burgers equations. The interaction between the nonlinear term and the dispersion term in the KdV-Burgers equation forms a stable forward-propagation soliton. Through mathematical analysis, the propagation path and speed of the soliton were described. By means of the singularly perturbed expansion method, the asymptotic solution to the problem was constructed. First, the degenerate solution was obtained with the Riemann-Earnshaw method, and the simple wave was obtained. There is a velocity difference between any point of the simple wave shape and the initial point, which makes the wave form continuously distorted in the process of propagation, and finally forms the shock wave surface, namely discontinuity. There is a time-varying jump in the velocity of particles between both sides of the discontinuity. Second, a modified traveling wave transformation was built through substitution of variables at the discontinuity of the degenerate solution, to obtain soliton solutions of the expansion of internal solutions and prove the existence and uniqueness of the internal and external solutions. Finally, the residual term was estimated with the existence of the uniformly bounded inverse operator, and the uniform effectiveness of the asymptotic solution was obtained. The results show that, the perturbations of KdV-Burgers equations with large Reynolds numbers and weak dispersions concentrate on the neighbourhoods of the discontinuities of the degenerate solutions. The soliton links the particles across the 2 sides, and its propagation path is not a linear form of time and space, but leads along the discontinuity of the degenerate solution, forming a stable waveform. |
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